Topological groups, $\mu$-types and their stabilizers

Abstract

We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M . To each such group G we associate a compact G -space of partial types, S^{\mu}_G(M)=\{p_{\mu}\colon p\in S_G(M)\} which is the quotient of the usual type space S_G(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab ^{\mu}(p) , which is the stabilizer of p_{\mu} . This group is nontrivial when p is unbounded in the sense of \mathcal M ; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of S^{\mu}_G(M) and its connection to the Samuel compactification of topological groups.